src/HOL/Univ.thy
 author paulson Thu Sep 23 13:06:31 1999 +0200 (1999-09-23) changeset 7584 5be4bb8e4e3f parent 7255 853bdbe9973d child 8735 bb2250ac9557 permissions -rw-r--r--
```     1 (*  Title:      HOL/Univ.thy
```
```     2     ID:         \$Id\$
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```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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```     4     Copyright   1993  University of Cambridge
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```     5
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```     6 Declares the type ('a, 'b) node, a subtype of (nat=>'b+nat) * ('a+nat)
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```     7
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```     8 Defines "Cartesian Product" and "Disjoint Sum" as set operations.
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```     9 Could <*> be generalized to a general summation (Sigma)?
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```    10 *)
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```    11
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```    12 Univ = Arith + Sum +
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```    13
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```    14 setup arith_setup
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```    15
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```    16
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```    17 (** lists, trees will be sets of nodes **)
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```    18
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```    19 global
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```    20
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```    21 typedef (Node)
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```    22   ('a, 'b) node = "{p. EX f x k. p = (f::nat=>'b+nat, x::'a+nat) & f k = Inr 0}"
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```    23
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```    24 types
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```    25   'a item = ('a, unit) node set
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```    26   ('a, 'b) dtree = ('a, 'b) node set
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```    27
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```    28 consts
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```    29   apfst     :: "['a=>'c, 'a*'b] => 'c*'b"
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```    30   Push      :: "[('b + nat), nat => ('b + nat)] => (nat => ('b + nat))"
```
```    31
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```    32   Push_Node :: "[('b + nat), ('a, 'b) node] => ('a, 'b) node"
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```    33   ndepth    :: ('a, 'b) node => nat
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```    34
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```    35   Atom      :: "('a + nat) => ('a, 'b) dtree"
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```    36   Leaf      :: 'a => ('a, 'b) dtree
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```    37   Numb      :: nat => ('a, 'b) dtree
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```    38   Scons     :: [('a, 'b) dtree, ('a, 'b) dtree] => ('a, 'b) dtree
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```    39   In0,In1   :: ('a, 'b) dtree => ('a, 'b) dtree
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```    40
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```    41   Lim       :: ('b => ('a, 'b) dtree) => ('a, 'b) dtree
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```    42   Funs      :: "'u set => ('t => 'u) set"
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```    43
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```    44   ntrunc    :: [nat, ('a, 'b) dtree] => ('a, 'b) dtree
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```    45
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```    46   uprod     :: [('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set
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```    47   usum      :: [('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set
```
```    48
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```    49   Split     :: [[('a, 'b) dtree, ('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c
```
```    50   Case      :: [[('a, 'b) dtree]=>'c, [('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c
```
```    51
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```    52   dprod     :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
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```    53                 => (('a, 'b) dtree * ('a, 'b) dtree)set"
```
```    54   dsum      :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
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```    55                 => (('a, 'b) dtree * ('a, 'b) dtree)set"
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```    56
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```    57
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```    58 local
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```    59
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```    60 defs
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```    61
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```    62   Push_Node_def  "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))"
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```    63
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```    64   (*crude "lists" of nats -- needed for the constructions*)
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```    65   apfst_def  "apfst == (%f (x,y). (f(x),y))"
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```    66   Push_def   "Push == (%b h. nat_case b h)"
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```    67
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```    68   (** operations on S-expressions -- sets of nodes **)
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```    69
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```    70   (*S-expression constructors*)
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```    71   Atom_def   "Atom == (%x. {Abs_Node((%k. Inr 0, x))})"
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```    72   Scons_def  "Scons M N == (Push_Node (Inr 1) `` M) Un (Push_Node (Inr 2) `` N)"
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```    73
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```    74   (*Leaf nodes, with arbitrary or nat labels*)
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```    75   Leaf_def   "Leaf == Atom o Inl"
```
```    76   Numb_def   "Numb == Atom o Inr"
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```    77
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```    78   (*Injections of the "disjoint sum"*)
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```    79   In0_def    "In0(M) == Scons (Numb 0) M"
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```    80   In1_def    "In1(M) == Scons (Numb 1) M"
```
```    81
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```    82   (*Function spaces*)
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```    83   Lim_def "Lim f == Union {z. ? x. z = Push_Node (Inl x) `` (f x)}"
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```    84   Funs_def "Funs S == {f. range f <= S}"
```
```    85
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```    86   (*the set of nodes with depth less than k*)
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```    87   ndepth_def "ndepth(n) == (%(f,x). LEAST k. f k = Inr 0) (Rep_Node n)"
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```    88   ntrunc_def "ntrunc k N == {n. n:N & ndepth(n)<k}"
```
```    89
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```    90   (*products and sums for the "universe"*)
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```    91   uprod_def  "uprod A B == UN x:A. UN y:B. { Scons x y }"
```
```    92   usum_def   "usum A B == In0``A Un In1``B"
```
```    93
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```    94   (*the corresponding eliminators*)
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```    95   Split_def  "Split c M == @u. ? x y. M = Scons x y & u = c x y"
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```    96
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```    97   Case_def   "Case c d M == @u.  (? x . M = In0(x) & u = c(x))
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```    98                                | (? y . M = In1(y) & u = d(y))"
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```    99
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```   100
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```   101   (** equality for the "universe" **)
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```   102
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```   103   dprod_def  "dprod r s == UN (x,x'):r. UN (y,y'):s. {(Scons x y, Scons x' y')}"
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```   104
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```   105   dsum_def   "dsum r s == (UN (x,x'):r. {(In0(x),In0(x'))}) Un
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```   106                           (UN (y,y'):s. {(In1(y),In1(y'))})"
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```   107
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```   108 end
```